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Salt water may have anti-septic properties due to the effect it has on water potential. Pure water has a water potential (Ψ) of zero. A concentrated salt solution has a lower (more-negative) water potential. The water potential of the salt solution is likely to be more negative than that of the pathogen's cytoplasm; the salt solution is therefore referred to as hypertonic. Therefore water osmoses out of the cell (osmosis being the net movement of water from a higher water potential to a lower water potential across a semi-permeable membrane). The loss of water from the pathogenic cells causes osmotic crenation - the cell becomes shrivelled and dies.

A hypotonic solution (for example cells placed into pure water) would cause the opposite effect - osmotic lysis. This is the bursting of the cell due to the movement of water into the cell. The bacterial cell wall would first have to be damaged (e.g. by penicillin). This would not be the process by which a salt solution has effect, however.

The fact that the salt water is warm in order to improve solubility may also have the side-effect of causing vasodilation around the infection, increasing the rate at which white blood cells can arrive at the infection site.

It has been more difficult to find a theory as to why a salt solution would have analgesic properties, see the comments below & previous versions of this answer.

It is my understanding that a sore throat is sore because of inflammation. I had always thought that the analgesic effect of salt water was to reduce that inflammation by placing the inflamed tissue in a hypertonic solution. Do you have a reference that says that the salt water is actually killing the pathogenic cells or that this is the direct mechanism of action for why it "helps" the symptoms of sore throat?
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yamadDec 19 '11 at 17:02

It would be doubtful that any antiseptic effect as I described above would produce immediate "help" or pain relief. I agree that the solution would have anti-inflamtory effect, however as to the "sore throat being sore because of the inflammation" I must confess that I don't know. If that is the case then of course that would be the direct mechanism that the op is looking for =)
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Rory MDec 19 '11 at 17:15

Well it certainly wouldn't discriminate; I would propose that your own cells are less isolated, in that they will be near to a capillary with ready supply of water to replace any lost in the osmosis?
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Rory MDec 27 '11 at 13:56

@Casebash It also helps to reduce inflammation by the same osmotic process - it draws the excess fluid around the cells thereby reducing the swelling. Reduced swelling would mean less irritation by the food/water entering the pharynx, as the contact surface is lessened.
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One FaceFeb 3 at 1:55